论文标题
光滑的涂鸦组，iii：平滑度定理
Smooth profinite groups, III: the Smoothness Theorem
论文作者
论文摘要
令$ p $为素数。在本文中，我们证明了平滑度定理，该定理断言$（1,1）$ - 循环对为$（n，1）$ - 环体元素，对于所有$ n \ geq 1 $。在Galois同胞学的特殊情况下，平滑度定理提供了一种新的证据，证明了规范残基同构定理，这完全是与动机共同体学的完全脱节。这种方法的副产品是，后者定理遵循单独的田野$ p^2 $ kummer理论。此外，我们将其扩展为从绝对的galois域组到代数曲线（不一定是光滑，也不是正确的）曲线的代数基本组。
Let $p$ be a prime. In this article, we prove the Smoothness Theorem, which asserts that a $(1,1)$-cyclotomic pair is $(n,1)$-cyclotomic, for all $n \geq 1$. In the particular case of Galois cohomology, the Smoothness Theorem provides a new proof of the Norm Residue Isomorphism Theorem, entirely disjoint from motivic cohomology. A byproduct of this approach, is that the latter Theorem follows from mod $p^2$ Kummer theory for fields alone. We moreover extend it, from absolute Galois groups of fields, to algebraic fundamental groups of (not necessarily smooth, nor proper) curves over algebraically closed fields.
